The Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance of the Work of Riemann, Christoffel and Ricci

The paper attempts to show how the Logical Empiricists' interpretation of the relation between geometry and reality emerges from a collision of mathematical traditions. Considering Riemann's work as the initiator of a 19th century geometrical tradition, whose main protagonists were Helmholtz and Poincaré, the Logical Empiricists neglected the fact that Riemann's revolutionary insight ourished rather in a non-geometrical tradition dominated by the works of Christo el and Ricci-Curbastro roughly in the same years. I will argue that in the attempt to draw the line Riemann-HelmholtzPoincaré-Einstein Logical Empiricists were led to argue that General Relativity raised mainly a problem of mathematical under-determination, i.e. the discovery that there are physical di erences that cannot be expressed in the relevant mathematical structure of the theory. However, a historical reconstruction of the alternative line of development Riemann-Chrito el-Ricci-Einstein shows on the contrary that the main philosophical issue raised by Einstein's theory was rather that of mathematical over-determination, i.e. the recognition of the presence of redundant mathematical di erences that do not have any correspondence in physical reality. In Logical Empiricists' philosophy of space and time, Einstein's conception of the relations between geometry and experience appears to be the heir of a 19th century philosophical and scienti c tradition, whose main protagonists were Riemann, Helmholtz, and Poincaré. The result of such a tradition appeared to be at best framed in Reichenbach's celebrated theory of equivalent descriptions . Riemann, Helmholtz and Poincaré discovered the principle of the relativity of geometry , i.e. our freedom to chose among di erent metric geometries in as much as they can be transformed into one another by unique and continuous transformations, that is in as much as they are, in the Logical Empiricists' parlance, topologically equivalent . It is probably the merit of Michael Friedman to have convincingly shown that such a tradition simply never existed (Friedman, 1995). Helmholtz's and Poincaré's philosophy of geometry presupposes homogenous spaces, which can be covered by congruent tiles without gaps and overlappings. In such geometries there is a unique set of congruence relations, on which all observer can agree, or, more technically, a group of self-mappings with the properties of rigid motions can be de ned. Riemann, on the contrary, left open the possibility of highly non-uniform spaces, where no group of motion can be de ned and thus no unique conventional agreement can be made as to which tiles are congruent. Logical Empiricists were of course aware of the elementary fact that there are no rigid bodies in spaces of variable curvature. However, by stripping Helmholtz and Poincaré philosophy of geometry from their group-theoretical implications (Friedman, 1995), they believed it was simply possible to shift the attention from nite rigid bodies to in nitesimal rigid rods . As Roberto Torretti has shown, however, this strategy is hardly compatible with conventionalism. In a Riemannian manifold, an in nitesimal rod is considered rigid as long as it has an Euclidean behavior; one